![Question ○ Use gradient to approximate|■8(sin(π/2+0.1)&ln(1.1)@3&√4)| ○ Let f(x,y,z)=|■8(sinx&lny@3&√z)|, then ○ f(π/2,1,4)=2 ○ Note ○ Near x=a, f(x)≈f(a)+∇f(a)(x−a) ○ Calculate Gradient ○ ∇f(x,y,z)=[█(f_x@f_y@f_z )]=[█(√z cosx@−3∕y@sinx∕(2√z) )] ○ ∇f(π/2,1,4)=[█(0@−3@1∕4)] ○ Approximation ○ f((π/2+0.1), (1+0.1), (4+0.1)) ○ ≈f(π/2,1,4)+∇f(π/2,1,4)(x−a) ○ =2+[█(0@−3@1∕4)]⋅[█(0.1@0.1@0.1)] ○ =2−0.3+0.025 ○ =1.725](https://shawnzhong.com/wp-content/uploads/2017/12/img_5a2b70ad62aaf.png)
Math 375 – 12/6
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